[Paper Review] Hydrodynamic limits and numerical errors of isothermal lattice Boltzmann schemes

Gauthier Wissocq, Pierre Sagaut|arXiv (Cornell University)|Apr 29, 2021

Lattice Boltzmann Simulation StudiesEngineering105 references27 citations

TL;DR

This paper introduces a novel asymptotic analysis based on Taylor expansion in the Knudsen number to systematically separate and identify consistency errors (from kinetic formulation) and numerical errors (from spatial and temporal discretization) in isothermal lattice Boltzmann schemes. It reveals that the low dissipation of the BGK model arises from a favorable error structure, while the regularized and MRT models suffer from 'hyperviscous degeneracy'—an artificial high-order dissipation caused by large numerical pre-factors, leading to over-dissipation and instability in low-viscosity flows.

ABSTRACT

With the aim of better understanding the numerical properties of the lattice Boltzmann method (LBM), a general methodology is proposed to derive its hydrodynamic limits in the discrete setting. It relies on a Taylor expansion in the limit of low Knudsen numbers. With a single asymptotic analysis, two kinds of deviations with the Navier-Stokes (NS) equations are explicitly evidenced: consistency errors, inherited from the kinetic description of the LBM, and numerical errors attributed to its space and time discretization. The methodology is applied to the Bhatnagar-Gross-Krook (BGK), the regularized and the multiple relaxation time (MRT) collision models in the isothermal framework. Deviation terms are systematically confronted to linear analyses in order to validate their expressions, interpret them and provide explanations for their numerical properties. The low dissipation of the BGK model is then related to a particular pattern of its error terms in the Taylor expansion. Similarly, dissipation properties of the regularized and MRT models are explained by a phenomenon referred to as hyperviscous degeneracy. The latter consists in an unexpected resurgence of high-order Knudsen effects induced by a large numerical pre-factor. It is at the origin of over-dissipation and severe instabilities in the low-viscosity regime.

Motivation & Objective

• To develop a unified methodology for deriving hydrodynamic limits of lattice Boltzmann schemes in the discrete setting.
• To explicitly separate consistency errors (from kinetic description) from numerical errors (from space/time discretization).
• To explain the numerical behavior of BGK, regularized, and MRT collision models through asymptotic analysis.
• To identify the origin of over-dissipation and instability in low-viscosity regimes for advanced LBM models.
• To validate error terms via linear stability analysis and provide physical interpretation of numerical properties.

Proposed method

• Applies a Taylor expansion in the Knudsen number to derive equivalent macroscopic equations from the discrete lattice Boltzmann equation.
• Performs a single asymptotic analysis without assuming a priori scaling between Knudsen number and discretization parameters (∆x, ∆t).
• Derives systematic expressions for consistency and numerical error terms up to fourth-order spatial derivatives.
• Compares derived error terms with results from linear stability analysis to validate and interpret their physical meaning.
• Analyzes the structure of error coefficients (C_i, N_i) for BGK, regularized, and MRT models to explain their dissipation properties.
• Identifies 'hyperviscous degeneracy' as a mechanism where large numerical pre-factors reintroduce high-order Knudsen effects, causing over-dissipation.

Experimental results

Research questions

• RQ1 How can consistency and numerical errors in lattice Boltzmann schemes be systematically separated and quantified in the discrete setting?
• RQ2 What causes the low dissipation observed in the BGK model, and how is it related to the structure of its error terms?
• RQ3 Why do regularized and MRT lattice Boltzmann models exhibit over-dissipation and instability in low-viscosity flows?
• RQ4 What is the role of the numerical pre-factor in the emergence of spurious high-order effects in MRT and regularized schemes?
• RQ5 How do the derived error terms compare with those from linear stability analysis, and what does this imply for model robustness?

Key findings

• The BGK model exhibits low dissipation due to a specific cancellation pattern in its error terms, particularly in the C1 and C2 coefficients.
• The regularized and MRT models suffer from 'hyperviscous degeneracy', where large numerical pre-factors reintroduce high-order Knudsen effects, leading to over-dissipation.
• This hyperviscous degeneracy is identified as the root cause of severe instabilities in the low-viscosity regime, especially for MRT and regularized schemes.
• The numerical error terms (N_i) are found to be significantly larger in magnitude than consistency errors (C_i) for MRT and regularized models, explaining their poor performance at low viscosity.
• The analysis validates that the error structure of the BGK model is more favorable than that of MRT and regularized models, explaining its better stability in low-viscosity flows.
• The methodology successfully distinguishes between physical consistency errors and purely numerical artifacts, enabling a clearer understanding of LBM's numerical properties.